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On dispersive equations and their
importance in mathematics
Gigliola StaffilaniRadcliffe Institute for Advanced Study and MIT
April 8, 2010
1 What is a dispersive equation
2 The KdV equation
3 The Schrodinger equation
4 Weak Turbulence
5 Other topics on Dispersive equations
What is a dispersive equation
The simplest possible evolution partial differential equation that isnot either hyperbolic or parabolic is the Airy Equation, whichinitial value problem (IVP) can be written as{
(−i∂t + ∂xxx)u = 0u(0, x) = u0(x),
where x ∈ R. The wave solution of this IVP is the simplestexample of a solution to a dispersive equation. We will computethis solution explicitly and we will see that it satisfies the following,rather informal, definition:
Definition
An evolution partial differential equation is dispersive if, when noboundary conditions are imposed, its wave solutions spread out inspace as they evolve in time.
To find a solution for the Airy Cauchy problem let’s look forplane-wave solutions: for fixed A and k we write
vk(x , t) = Ae(kx−ωt) = Aek(x−ω/kt)
where k =wave number and ω =angular frequency. If wesubstitute vk into our equation we obtain the relationship
Aek(x−ω/kt)[−iω + (ik)3] = 0
and from here
ω =(ik)3
i⇐⇒ ω
k= −k2.
The equationω
k= −k2
is called the Dispersive Relation for the Airy equation.
Remark
The dispersive relations says that “plane waves with large wavenumber travel faster than those with a smaller one”. This is thereason why there is “spreading”. In mathematical terms thisphenomenon is called broadening of the wave packet.
To understand this better let’s use the Fourier transform: for theinitial data we have
u0(x) =
∫u0(k)e ikxdk
Now think of∫
as a sum and, for fixed k, of vk(x) = u0(k)e ikx asa wave. Then each wave vk(x) evolves into
vk(x , t) = u0(k)e ik(x+tk2),
where the wave with larger k travels faster.
By “adding up” all these waves we obtain the solution to the AiryIVP
u(x , t) =
∫u0(k)e ik(x+k2t)dk.
(We will see that in the periodic case this will be different sincethere boundary conditions are imposed!) It is instructive tocontrast the Airy equation with the transport equation{
(∂t + C∂x)u = 0u(0, x) = u0(x),
where x ∈ R, and let’s pick C > 0. The dispersive relation isωk = C , that is the velocity is constant, so the wave packet travelswith the same speed and there is no dispersion.
Definition (More formal)
We say that an evolution equation (defined on Rn), is dispersive ifits dispersive relation ω(k)/|k| = g(k) is a real function such that
|g(k)| → ∞ as |k| → ∞.
There is also a more geometric/analytic definition of dispersion.Back to the solution u of the Airy IVP, we recall that
u(x , t) =: W (t)u0(x) =
∫u0(k)e i(xk+k3t)dk.
If we define the curve
S = {(k, τ)/τ = k3, k ∈ R},then one can also write
u(x , t) =: W (t)u0(x) = R∗u0(x , t),
where R∗ is the adjoint of R, the Fourier restriction operator on S :
R(f ) := f (k, k2).
Remark
Since in general f belongs only to L2(R× R) and S is of Lebesguemeasure zero, it is not obvious that f restricted to S even makessense!
Theorem (Informal)
If S is a “curved” graph then for any f ∈ L2(R× R) its restrictionon S is well defined and moreover “good” estimates can be proved.
See for example Harmonic Analysis by E. Stein.
Examples of Dispersive Equations
• The (generalized) KdV equation:
∂tu + ∂xxxu + γuk∂xu = 0,
• Nonlinear Schrodinger equation:
i∂tu + ∆u + N(u,Du) = 0,
• Boussinesq equation:
∂ttu − ∂xxu − ∂xx(1/2u2 + ∂xxu) = 0,
These equations were all introduced in order to describe a certainwave phenomena. As a consequence the first obvious questionsthat one would like to address are: existence, uniqueness andstability of solutions (local well-posedness), maximum time ofexistence, blow up, scattering, existence of solitons etc.It turned out that while investigating these questions ones stepsout of the field of harmonic or Fourier analysis and enters othersfields like symplectic geometry, analytic number theory, probabilityand dynamical systems.To illustrate these interactions I will first look at KdV typeequations and then at Schrodinger ones.
The KdV equation
The (generalized) KdV initial value problem takes the form of{∂tu + ∂xxxu + uk∂xu = 0
u(0, x) = u0(x),
This problem models long waves along a shallow channel. Linkedto this problem is the discovery of solitons: In 1834 a navalarchitect John Scott Russel, while riding his horse along a canalobserved the first recorded soliton. The “event” was repeated in a“controlled” manner in 1995:
Definition
A soliton is a self-reinforcing solitary wave (a wave packet or pulse)that maintains its shape while it travels at a constant speed.
Remark
Clearly a soliton cannot be solution to the Airy equation (rememberdispersion!!). In fact solitons are caused by a perfect cancellationof nonlinear and dispersive effects in the medium. An interestingfeature of solitons is that in spite of the fact that they arenonlinear phenomena, they behave “linearly” when they interact!
We actually have an explicit form of solitons to a (generalized)KdV equation:
uc,k(x , t) = Φc,k(x − ct), for c > 0
where Φc,k = c(k + 2)/2 sech2(k/2√
cx)1/k .
Interaction of two solitons
KdV as an integrable system
Consider the equation (k=1)
∂tu + ∂xxxu + u∂xu = 0.
Fermi, Pasta and Ulam used this equation to justify the apparentparadox in chaos theory that many “complicated enough” physicalsystems exhibit almost exactly periodic behavior instead of ergodicbehavior! There are many (equivalent) ways of formulating thisfact:
• The system is integrable
• The equation admits Lax pairs
• Inverse scattering completely solves the IVP.
• The system admits infinitely many conservation laws
Some conservation laws
The first 3 for the KdV equation are:∫u(x , t) dx∫u2(x , t) dx mass∫u2x (x , t)− u3
3(x , t) ds Hamiltonian
The first complete algorithm to compute all the conservation lawsis due to Miura, Gardner and Kruskal. In principle at this point wedon’t even know if the integrals above actually are finite, we don’tknow yet if the solution u(x , t) exist!
Well-posedness for (generalized) KdV
We consider the IVP{∂tu + ∂xxxu + uk∂xu = 0
u(0, x) = u0(x),
We introduce the Sobolev space Hs by recalling that the norm of afunction f in this space is(∫
|f (k)|2(1 + |k|)2s dk
)1/2
< ∞
Definition
The IVP is locally well-posed in Hs if for any u0 ∈ Hs there existsT = T (u0) and a unique solution u in a Banach spaceX s
T ⊂ C ([0,T ],Hs). Moreover there is continuity with respect tothe initial data. If T can be taken arbitrerely large we say that wehave global well-posed.
The “classical” method to prove well-posedness for dispersiveequations is by a priori estimates. This is the Energy Method.These a priori bounds are found by using the equation andintegration by parts. Using this methods the KdV equation can beproved to be locally well posed in Hs , s > 3/2.The second method, developed by Kenig, Ponce and Vega isbased on Oscillatory Integrals. To understand this method we firstobserve that by the Duhamel Principle one can rewrite the IVP asan “equivalent” integral equation:
u(x , t) = W (t)u0(x) + c
∫ t
0W (t − t ′)(u∂xu)(x , t ′) dt ′,
where, as we know, W (t)u0(x) = R∗(u0)(x , t) is the solution ofthe linear IVP.
This methods is based on the following steps:
• The proof of several estimates for W (t)u0(x) using theFourier restriction operator R and other estimates based onoscillatory integrals.
• The definition of a Banach space X sT of space time functions
where the norms of the above estimates are considered.• The use of the Banach space X s
T ⊂ C ([0,T ],Hs) as a spacewhere to look for the fixed point of the operator
Lv(x , t) = W (t)u0(x) + c
∫ t
0W (t − t ′)(v∂xv)(x , t ′) dt ′.
• The use of the integral equation above to claim that the fixedpoint is the unique solution of the equation.
This method allowed Kenig, Ponce and Vega to improve localwell-posedness for the KdV IVP to Hs , s > 3/4, and globalwell-posedness in H1. Similar results for the generalized (k > 1)KdV equations.
Bourgain’s Contribution
Bourgain continues with the fixed point idea, but introduces in thiscontext another type of space: X s,b. The norm of a functionf ∈ X s,b is defined by
‖f ‖X s,b =
(∫∫|f (k, τ)|2 < k >2s< τ − k3 >2b dk dτ
)1/2
.
Again we see reappearing the cubic
S = {(k, τ)/τ = k3, k ∈ R}.
With this space Bourgain was able to now attack also the periodicKdV equation. Here oscillatory integrals cannot be used sinceForier transform gives oscillatory series, not integrals! These spaceswere also later used by Kenig, Ponce and Vega to prove local wellposedness for negative Sobolev spaces H−ρ. More precisely on theline ρ < 3/4 and on the circle ρ ≤ 1/2.
The symplectic KdV flow
Why do we care about negative Sobolev spaces? One very goodreason is described below. If we consider the periodic case and wetake Forier transform in space of the solution u
u(x , t) ⇐⇒ (u(k, t))k∈Z,
we can view the IVP as an Hamiltonian system of infinitedimension for the infinite vector (u(k))k∈Z. For this system we candefine the symplectic form
(u, v) =
∫u∂−1
x v dx
on the Sobolev space H−1/2. It is then reasonable to ask if certaintheorems (see Gromov) that are proved in the finite dimensionsetting are still true here.
Theorem
(Colliander, Keel, S, Takaoka and Tao) The symplectic KdV flow isglobal in time on H−1/2 and the Gromov non-squeezing theoremholds.
This theorem has two major parts. The first deals with extendinglocal well-posedness in H−1/2 to global well-posedness by the useof the I-Method. The second part deals with proving thenon-squeezing theorem by approximating the system with a finitedimension one in an appropriate way and then by taking the limit.Similar results where obtained earlier by Kuksin for compactperturbation of certain linear systems and by Bourgain for a certainSchrodinger equation.
The Schrodinger equation
The Schrodinger equation describes for example how quantumstates of a physical system change in time. One example is the IVP
NLS
{i∂tu + ∆u + σ|u|p−1u = 0u(0, x) = u0(x), x ∈ Rn
with p > 1 and σ = ±1.The solution of its linear IVP is
S(t)u0(x) =
∫e i(x ·k+t|k|2)u0(k) dk = R∗(u0),
where now R(f ) = f (k, |k|2), the operator that restrict the Fouriertransform on the surface given by
P = {(k, τ)/τ = |k|2, k ∈ Rn}.
The Strichartz estimates in Rn
In order to prove (local) well-posedness for Schrodinger equationsthe Strichartz Estimates are fundamental.We call admissible couple any pare of exponents (q, r) such that
2
q= n
(1
2− 1
r
)q > 2, r < ∞.
Then for any admissible couple (q, r)
‖S(t)u0‖Lqt L
rx
. ‖u0‖L2x,
and for any other admissible couple (q, r)∥∥∥∥∫ t
0S(t − t ′)F (t ′) dt ′
∥∥∥∥Lq
t Lrx
. ‖F‖Lq′
t Lr′x.
Thanks to these estimates one can then prove well-posednessresults for Schrodinger type IVP via fixed point theorems.
How difficult is it to prove well-posedness? It is certainly easier if
• the interval of time [0,T ] is short,
• the initial data are small,
• the nonlinearity is weak.
In fact, from the Duhamel principle u is solution to the NLSequation above if and only if
u(x , t) = S(t)u0 + c
∫ t
0S(t − t ′)σ|u|p−1u(t ′) dt ′
and if one could claim that the non-linear perm is a “small”perturbation, then a fixed point theorem in the space of theStrichartz norms will provide well-posedness, at least for shorttimes. The question of long time well-posedness or blow up is farmore complex.
Scaling
There are many important player in the game of well-posedness forNLS on Rn: scaling invariance and conservation laws, monotonicityformula (i.e. Morawetz type estimates), Viriel identities and otherkind of symmetries. Here we only consider the first one as anexample: if u solves NLS above then
uλ(x , t) = λ− 2
p−1 u(x
λ,
t
λ2
)solves the same equation with initial datum u0,λ = u0(
xλ). We have
that‖u0,λ‖Hs ∼ λsc−s‖u0‖Hs
where sc = n/2− 2/(p − 1) is the critical exponent. We can now“classify” the difficulty of the NLS problem above in terms of sc .
So for sc = n/2− 2/(p − 1) and since ‖u0,λ‖Hs ∼ λsc−s , we have
• If s < sc the space Hs is supercritical(as λ →∞ the norm of ‖u0,λ‖Hs grows)
• If s = sc the space Hs is critical(as λ →∞ the norm of ‖u0,λ‖Hs does not change)
• If s > sc the space Hs is subcritical(as λ →∞ the norm of ‖u0,λ‖Hs gets smaller).
Local well-posedness is by now very well understood, (see alsoCazenave, Weissler, Kato, Tsutsumi, Ginibre,Velo etc). In recentyears a lot of progress as been made to prove global well-posednessat the level of the energy (H1) or mass (L2) norms, even whenthey are “critical spaces” in the sense above.
Energy critical NLS in R3
These are two types of theorems now available:
Theorem (Defocusing case σ = −1)
Assume that the energy of the quintic defocusing NLS∫Rn
1
2|∇u|2 dx − 2σ
6
∫Rn
|u|6 dx
is finite. Then the IVP is globally well-posed and at infinity (intime) the solution approximate a linear one (scattering).
For the proof see Bourgain and Grillakis in the radial case,Colliander-Keel-S-Takaoka-Tao for the general case. Also seeRyckman-Visan for n = 4 and Visan for n > 4.
Theorem (Focusing case σ = 1)
Assume that u is the solution of the quintic focusing NLS and
supt‖∇u0(t)‖L2
x< ‖∇W ‖L2
x,
where W is the stationary solution. Then if we also assume radialsymmetry the same conclusion as above holds.
For the proof see Kenig-Merle, See also Killip-Visan for n ≥ 5,where the radial assumption has been removed.In a certain sense the complement of this last theorem is acollection of recent and strong results of blow up rate and blow upprofile due to Merle-Raphael.
Schrodinger equations on manifolds
Given a manifold (M, g) equipped with its Laplace-Beltramioperator ∆g one can certainly define a Schrodinger equation on it.The interesting questions here are related to the understanding ofthe influence of the geometry on the behavior of thesolutions...when they exist. I will list some results by concentratingon Strichartz estimates:
• On the sphere Sn: There is a loss of 1n derivative.
Burq-Gerard-Tzvetkov
• On the hyperbolic space Hn: There is a larger family ofStrichartz estimates.Banica, Banica-Carles-S, Ionescu-S, Banica-Carles-Duyckaerts,Ancher-Pierfelice
• On the torus Tn: Limited Strichartz estimates.Bourgain
Here Tn is the Torus for which the symbol of ∆ is∑n
i=1 k2i !
The cubic defocusing Schrodinger equation in
T2
Well-posedness in Hs , s > 0 follows from the Strichartz estimatedue to Bourgain:
‖S(t)u0‖L4TL4
T≤ ‖u0‖Hε
x
for any ε > 0. The interesting part of the proof of this estimate isthat it is based on counting the lattice points on a “thin” sphere
k21 + k2
2 = R2 + θ, |θ| < 1,R >> 1.
Here Gauss lemma is used to claim that this number is smallerthan Rε for any ε > 0.What about irrational tori? If one wants to use the same methodof Bourgain then one needs to have a good estimate for thenumber of lattice points on a “thin” ellipsoid. This is basically anopen question except for a partial result of Bourgain.
Notion of Weak Turbulence
Definition
Weak turbulence is the phenomenon of global-in-time solutionsshifting their mass toward increasingly high frequencies.
This shift is also called forward cascade.
One way of measuring weak turbulence is to consider thefunction in time
‖u(t)‖2Hs =
∫|u(t, k)|2|k|2sdξ
for s � 1 and prove that it grows for large times t.
Weak turbulence is incompatible with scattering or completeintegrability.
Conjecture
Solutions to dispersive equations on Rn DO NOT exhibit weakturbulence. There are solutions to dispersive equations on Tn thatexhibit weak turbulence. In particular for NLS(T2) there existsu(x , t) s. t. ‖u(t)‖2
Hs →∞ as t →∞.
Theorem (Colliander-Keel-Staffilani-Takaoka-Tao)
Let s > 1, K � 1 and 0 < σ < 1 be given. Then there exist aglobal smooth solution u(x , t) to the defocusing IVP{
(i∂t + ∆)u = −|u|2uu(0, x) = u0(x), where x ∈ T2,
(NLS(T2))
and T > 0 such that ‖u0‖Hs ≤ σ and ‖u(T )‖2Hs ≥ K .
The toy model
The idea of the proof is to make the ansatz
v(t, x) =∑n∈Z2
an(t)ei(〈n,x〉+|n|2t),
and rewrite the equation as an ODE in terms of the infinite vector(an(t)). We also consider only the resonant part of the ODE andwe construct a special finite set of frequencies Λ that is closedunder resonance and has several other “good”properties. Thanksto these properties we arrive to a finite dimension toy model
−i∂tbj(t) = −bj(t)|bj(t)|2 − 2bj−1(t)2bj(t)− 2bj+1(t)
2bj(t),
for j = 0, ....,M + 1, with the boundary condition
b0(t) = bM+1(t) = 0.
Remark
This new IVP conserves the momentum, the mass(∑M
j=1 |bj(t)|2 = 1) and the energy!
Global well-posedness for this system is not an issue. Then wedefine
Σ = {x ∈ CM / |x |2 = 1} and W (t) : Σ → Σ,
where W (t)b(0) = b(t) for any solution b(t) of our system. It iseasy to see that if we define the torus
Tj = {(b1, ...., bM) ∈ Σ / |bj | = 1, bk = 0, k 6= j}
thenW (t)Tj = Tj for all j = 1, ....,M
(Tj is invariant).
At this point the problem has been set up in such a way that if wecould show that once we start “near” one of the first tori (lowfrequencies) we end up at a certain time T near one of the last tori(high frequencies) then we are done. In fact we have the followingresult:
Theorem
Let M ≥ 6. Given ε > 0 there exist x3 within ε of T3 and xM−2
within ε of TM−2 and a time T such that
W (T )x3 = xM−2.
Remark
Our theorem does not show that one can find a solution u whichHs norm grows in time. We cannot even prove that it grown as alog |t|!
I would like to conclude by listing other topics of great interest andintense mathematical activity:
• Wave mapsNahmod-Stefanov-Uhlenbeck, Klainerman-Rodnianski,Krieger-Schlag, Shatah-Struwe, Sterbenz-Tataru, Tao.
• Schrodinger mapsBejenaru-Kenig-Ionescu, Chang-Shatah-Uhlenbeck,Ding-Wang, Kenig-Lamm-Pollack-S-Toro, McGahagan,Nahmod-Stefanov-Uhlenbeck, Rodnianski-Rubenstain-S,Terng-Uhlenbeck.
• “Almost surely” well-posednessOh, Bourgain, Burq-Tzvetkov, Oh-Rey Bellet-Nahmod-S
• Regularity theorems for supercritical dispersive equations(similar to Navier-Stokes)
